The APA has published "Report of the American Psychological Association Task Force on Appropriate Therapeutic Responses to Sexual Orientation". Among it's findings, it says ""sexual orientation identity—not sexual orientation—appears to change via psychotherapy, support groups, and life events."
Support groups can include ex-gay groups. The report also says:
"For instance, participants reported benefits from mutual support groups, both sexual-minority affirming and ex-gay groups."
So while it says that sexual orientation identity can change through ex-gay groups, it does not say that sexual orientation can change through ex-gay groups or through therapy. I agree, from my experience, sexual orientation can only be changed through Jesus Christ.
I don't think the APA findings conflict with the teachings of the church at all. In fact, some of the findings seem to confirm what the church has been teaching since 1964. When Spencer W. Kimball talked about people who were gay who later became straight, he did not say the attractions would go away, but that is was "like the cure for alcoholism subject to continued vigilance." In every AA meeting they get together and say they are an alcoholic, even if they have been sober for decades, because the lure is still there. The APA report confirms what President Kimball said back in 1964 that the identity can change, but that the attractions may persist. Nice for the APA to finally catch up.
That isn't to say I think sexual orientation cannot change. The church teaches "While many Latter-day Saints, through individual effort, the exercise of faith, and reliance upon the enabling power of the Atonement, overcome same-gender attraction in mortality, others may not be free of this challenge in this life."(God Loveth His Children) Note the difference between the APA statement that sexual orientation cannot be changed through therapy, and the church's testimony that same-gender attraction can be overcome through the enabling power of the Atonement. The APA study did not research the enabling power of the atonement.
However, the APA did look at the affects of religion and stated:
"For instance, several publications indicate that active engagement with religious texts can reduce identity conflicts by reducing the salience of negative messages about homosexuality and increasing self-authority or understanding."
They did not reject the idea of celibacy for some clients:
"Some religious individuals may wish to resolve the tension between values and sexual orientation by choosing celibacy (sexual abstinence), which in some faiths, but not all, may be a virtuous path. We found limited empirical research on the mental health consequences of that course of action. Some clinical articles and surveys of individuals indicate that some may find such a life fulfilling; however, there are others who cannot achieve such a goal and might struggle with depression and loneliness. In a similar way, acting on same-sex sexual attractions may not be fulfilling solutions for others. Licensed mental health providers may approach such a situation by neither rejecting nor promoting celibacy."
The statement that sexual orientation cannot be changed by therapy is not a statement that the person cannot find fulfillment in celibacy. It also talks about how a counselor with a married client should not dismiss the possibility of saving the marriage. It recognized the importance of a person's religious identity, which may include a command for chastity, and confirmed that it does not need to conflict with our sexual orientation identity:
"Identities can be explored, experienced, or integrated without privileging or surrendering one or another at any age."
Our leaders have not denied the existence of a sexual orientation, but have only emphasized that it is subordinate to our identity as a child or God:
"There’s no denial that one’s gender orientation is certainly a core characteristic of any person, but it’s not the only one." (Elder Oaks)
I see no conflict between the findings of the APA and the teachings of the Church of Jesus Christ of Latter-day Saints. Much of the confusion stems from using different vocabulary than the rest of the world. If being gay means living the gay lifestyle for us, but it means having same-sex attractions for others, then it is going to create problems.